Fragments of the theory of the enumeration degrees

Abstract

We prove that every finite distributive lattice can be strongly embedded into the enumeration degrees as an interval, i.e., that there is an interval [a0,a1] of enumeration degrees isomorphic to the lattice, and any enumeration degree b≤a1 lies in this interval or below a0. As corollaries, we conclude that the ∃∀∃-theory of De is undecidable, while the extension of embeddings problem (a subproblem of the ∀∃-theory) is decidable.